Nonunitary transformations in quantum optics

Volume 137, number 7,8

PHYSICS LETTERS A

29 May 1989

NONUNITARY TRANSFORMATIONS IN QUANTUM OPTICS D.N. KLYSHKO Department ofPhysics, Moscow University, Moscow 119 899, USSR Received 1 March 1989; accepted forpublication 22 March 1989 Communicated by V.M. Agranovich

A general method ofquantum description of linear optical devices — beam splitters, lenses, etc. — with dissipation is discussed. The method is grounded by means of a kinetic equation. As an example the transformation ofthe field’s statistics by an absorbing half-transparent mirror is considered. It is shown that a heated mirror radiates correlated signals into pairs of mirror-conjugated modes.

1. Over the last two years there seems to be a renewed interest in the quantum theory of spectral fluters andbeam splitters [1—5],caused mainly by their possible use in experiments involving squeezed states (see ref. [2]), photon antibunching [1,3,4], quantum nondemolition measurement [61, the EPR-paradox [7] and other essentially nonclassical optical effects. The various methods, used in refs. [1—51, have a common feature: they are applicable only to dissipationless devices, which can be described by unitary transformations. But in real devices there is always some dissipation small in a dielectric mirror or a Nicol prism and deliberately large in a coloured glass or a spacefrequency filter. Dispersion of the refractive index, used for spectral filtration, is necessarily accompanied by absorption. The transformation of the field’s statistics by a photon counter or a heterodyne detector is also nonunitary as the detector’s quanturn yield i~is less than 1. This is also the case if in an ideal beam splitter or other dissipationless device some modes are not taken into account. A quantum amplifier has negative absorption (~>1) and hence should also be described by a nonunitary transformation (an active interferometer with amplification was considered in ref. [81).

dissipation: time- and space-frequency filters, lenses, interferometers, beam splitters, heterodyne detectors, etc. All these “filters” change the input field’s statistics and the task is to define this change through a set of phenomenological parameters of the filters. A convenient method is based on the division of the radiation field, surrounding the filter, into input and output parts and on the notion of a “transformation matrix” (TM) Dk/, connecting the Fourier components of these parts [91: b~=

~ Dk

1a7,

bk=

~ D~1a1,

(1)

—

—

2. In the present communication a general formalism is considered, which is applicable to the description of various optical devices with or without 334

or in vector notation b + = D a b D* a. Here a a (b b) are the creation and annihilation operators of the input (output) field, indices k, 1 include the wave vectors k, I and the polarisation indices Uk, V1. As a,~,bk are supposed to be time independent, so eq. (1) is applicable only to free fields (there is no such limitation in ref. [1]). The linearity of the fluter implies that the TM is diagonal in 1k I = (Ok/C, i.e. Dkl ö (w~ w,) (here continuous mode distribution is implicated), and thus the summation or integration in (1) is actually performed only over various directions of! and over v,. In case of a beam splitter the number of interacting modes n is 2, in case of a lense n=~. The TM of a dissipationless filter is unitary: D- D~= I, so that the “absorption matrix” A = I D• D~ is zero and the transformation (1) ~,

~,

‘-~

—

—

0375-9601/89/s 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Volume 137, number 7,8

PHYSICS LETTERS A

conserves the commutation relations. Note that it is possible to define D and A through the commutators: D= [b~, a], A=I+ [b~, b}. (2) —

It is clear that Dk/ plays the role of propagator from channel k to / (the free space propagators exp (ikr) should be added if necessary). In addition the TM determines the filter’s own thermal radiation via a

29 May 1989

fore averaging (over the input state) and in certain cases the chaotic radiation, connected with the absorption or amplification, should be taken into account. As will be shown below, both these demands are satisfied, if the following characteristic function is used:

x0~1(p)=x,~(b~u)x~(p)

(6)

,

Kirchhoff-type relation [9]. The relation (1) may be presented as the similaritytransformation in the Hubert space: bk= U ‘akU. If n=2, then U has the following structure [3]:

where

U= exp(iaa j~a1

i,,, (p) is the characteristic function of the in-field

+ ifia~a2)

Xexp(iyata2 +iôa~a1).

(3)

Comparison with (1) yields D’~’1=eia cos( W), D’~2=iyeia sinc( ~ D~1= iôe’~sinc ( hJ~), D~2= e’~ cos ( v’), (4) 2= yd and sinc(x) = x —‘ sin (x). The inverse where ~Pare relations

x~=e,~p(_p.~~.p*), A~=N(T~A,

N~T~ =dkIN~(wk)

=ôkI[exp(hwk/kT)—

Dkl=Dlk,

l]~

(the second argument ~ is being omitted for brevity), xTCU) determines the filter’s own radiation (T is the temperature of the absorbing elements, supposed to be homogeneous). This radiation “jams” the input information and nonclassical features (in addition to the destructive influence of the mode mixing and stochastic absorption itself see eq. (15) below) and plays a principal role in active filters with T<0 [8]. —

2ia = in (DD 2ô=

—

11 ID22 )*, 2ifl=in (DD22/D11 )*, C cos~(B), (5)

C where y=

2, D=Det(D),

B*= (D11D22/D)”

C”= (D12 D22/D21 D11 ) 1/2~ If n>2, then U can be expressed as a combination of two-mode operators (3). 4=Ifollows UU~=I(note the differentFrom senseD~D of the symbol “+“ in these equations), so that U has the meaning of the usual evolution operator [2,3]. Now in (4), (5) d11 = d22, d12 = d21 = (1 —d11) 1/2

Quantum optical systems usually are loaded by one or more detectors. Multiplying D by the diagonal matrix of the detector’s efficiency, DJ~”= ôkI?1 k’ , we get the total TM D 10~=D~°~-D, which takes account of the effective absorption in the detectors. Now the function Xtot Cu) =x 0~1( D~°~ -p) defines the statistics of the really observed electric signals. If the in-field is chaotic, i.e. if ~1~(p)=exp(_p.N.p*), Nk,= ~fl, (7) then according to (6) it retains its character (with modified N, which is defined in eq. (10) below). The Fourier transformation of (6) gives the output Glauber—Sudarshan representation P 0~ (ex) as the convolution of P1,, (~)with the Gaussian function: POUl(fl)=[1t”Det(AT)J~

(here dk,= lDk,I, 9~k/=arg(Dk,)),so that a, are real:

fi and

~“

a=—~’11, P~22, y=ô”’=iWexp(i~ 11—i~,12), O~P~it/2

x

J

P~(~) exp[

—

(D*.~_fl) 2ak.

.Afl(D.z*....$*)]

ft d

(8)

(essentially the same result is given in ref. [3fl

The derivatives of (6) at a = 0 determine the normally ordered output moments, for example,

3. IfA,&O then b~,b should be normally ordered be-

0~1= ,~=D*. 1~,

(9) 335

Volume 137, number 7,8

PHYSICS LETTERSA

N’=D-N-D~+A~.

(10)

According to (9) the TM can be measured by means of a coherent field, i.e. it can be defined operationally. Eq. (10) with N=O resemblesthe Kirchhofflaw: its diagonal elements express the thermal radiation intensity N kkm N ~ through temperature T and the filter’s “absorptivity” Ak,. (for the backward mode k [91); the nondiagonal elements Aki determine the correlation between modes k and / (see example below). Let N = 0 in (10). As the matrix N’ is nonnegative by definition, so a passive filter should have a nonnegative absorption matrix A. Hence

29 May 1989

With k= 1 eq. (13) describes the redistribution of the input photon noise between output modes (cf. refs. [2—5]): —2) , (14) g~’—2= ~ e~1(g~21 / where

—

‘2 , Akk~O, Ak,.A/l>~IA,./I

(11)

...

(in active filters the inequalities have the opposite sense). Let us find from (10) the reduction of the total number of photons in a cold filter: ~

—

~

= ID,.,

I 2N

2Nm

)

—

~ I D~mI If all in-modes are in the independent photonnumber states IN 2)= 1 NI’ and the 1>, then g~ Fano factor Fm AN2/N= 1 +N(g’21— 1) is zero (perfect photon antibunching). From (14) 1(

—

F’k = 1 +N~( 1

—

~ Ck,( 1 +NI’)).

(15)

Thus even in case of a noiseless input the output still acquires some noise, F> 0— as a result ofmixing and

= Sp(N — N’)

=
absorption. If only one input mode is excited, N 5jcaNa, then Ck, = öla and F ~ = 1 I D,.,~12 From 1 =~ (13) now follows g,~~)’ =g~. The same conversation law holds for all higher-order moments: the input normalized factorial moment g,(f~,is replicated —

This implies that the matrix I nonnegative.

—

fIb + should also be

4. In case of a cold or dissipationless filter A~=0 and in (6) XTCU) =1. As a result the normally ordered observables f~,the characteristic functions x and the P-functions only change their scales:

at all outputs (this is a generalisation of the well known one-mode rule). In a homodyne detector of squeezed states the observed electric signals can be expressed through the fourth normally ordered moment at the output of a

X

0~~Cu) =x~(b~u) ,

~

(l2a) D”-a))1~,

P0~,(~) = [Det(D-D~)

1 ‘P~(D”’

‘•~)

(12b)

beam splitter (see ref. [2]) and therefore eq. (l2b) can be used directly. As a result follows essentially the same expression as in ref. [2], but without the

(12c)

restriction ofunitarity.

(cf. refs. [2,5]). Consider the fourth moments as the output of a cold filter. In case of independent incoherent inputs we find from (1 2b) ~ e~,,,,(g~,,~—2) , where Nk/ 2’ gj~) = NkN, (NkN,)” I1)km1)/mINm Ck/m= (N’,.N~)”2

336

(13)

5. As a specific example consider a beam splitter in the form of a half-transparent absorbing mirror. Such a mirror couples the modes 1 and 2 with opposite signs of k~,so that A,2 ,~0 and the mirror’s own thermal radiation in these modes should be correlated. Let the TM have the following symmetry: D1, = D22 t”’, D,2 = D2, r* (we ignore the polarization), then A12 is real. From (10) with N=0 follows g~j’= —2 Re(rt*)/A, Am 1— T—R TmItI2,

RmIrI2.

(16)

Volume 137, number 7,8

PHYSICS LETTERS A

The intensity correlation is described by (13) with = 2. Note that r and t are limited by (11). According to the Airy formulae [10] for a plane layer in case k~ = k~ = 0, r,,—r,~’ r0t0 —r5~~

29 May 1989

with ic< 0 describe the correlation in the output noise of a quantum amplifier with a Fabry—Perot resonator. In case of small absorption or amplification (I KI n, icz ~ 1) g~) + 1 if nz= 2kE and g”~—l ifnz=(2k+1)x. .~

____________ ‘

r= r010 —r,j’ ~

(17)

where

The redistribution of noise by a cold mirror is and determined by (14) with e11=(1+RN2/TN1)’ 12=(1+TN1/RN2)* Eq. (15)nowtakestheform 2NI+R2N

1—fl r0=—, l+n

t,,=exp(iflz),

z=wh/c,

F~=l—

fl = n + ilc is the refractive index and h is the layer’s thickness. From (17) follows, that when h—~0,the numerator and the denominator in (16) are both equal to 2nKz, so that g ~)‘ —s 1. In fig. 1 is plotted the dependence of g( I), T, R and A on z for several values of n and K. Note, that even a practically nontransparent mirror with T= 10—i gives notable correlationg” =0.25 (z= 1 on fig. la). Eqs. (16), (17)

T

2—2TRNIN2

TNI+RN2

(18)

which is essentially the same as in ref. [3], the only difference being the absence of the unitary conditions, T+ R = 1, Re (rt*) = 0 (which are satisfied by (17) if K= 0). 6. As an example of a nonunitary filter with n = consider an ideal collecting lens with Gaussian apodisation, which has the following TM in the mixed (r, k)-representation [11]:

(a)

\~ \/

where D(r)=— ~exp{ik[r+~2(q/k_p/z)2]}, 1=f’—z’—i(kR2)’, p=(x,y),

A

0.5

1

_____ ________________________

1

_____

a.: ______________________

1

0

1

2

g

at r, created by a plane wave k, falling on the lens. of the lens (the origin r= 0 is at the centre of the lens). The function (19) has the meaning of the field In the limitf= ~, R =~ (19) approaches exp(ik-r). The transformation (19), which is unitary only in the limit R = 00, was used in ref. [11] for analysing some peculiarities in the space structure of g(2)’ in the case of focussed two-photon light, generated via parametric down conversion (various transformations of such light were also considered in refs. [7,121 in connection with the EPR-paradox).

(b)

A

-0.5

(19)

k~>0,and the effective radius fq=(k~,k~),z>0, and R are the focal length

\\~\

~

3

Fig. I. The correlation of thermal radiation in two mirror-conjugated modes: g is the first-order normalized correlator, z = coh/ c, h, T, R and A are the mirror’s thickness, transmissivity, reflectivity and absorptivity; the complex refractive index ñ is taken to be (a) 0.2+i3.44 (silver [101) and (b) 2.0+iO.2.

several methods [91: ~ngeven’s, Nyquist’sand othThe basicfor equation by ers.7. Consider example(6) M may atomsbeingrounded equilibrium states with a common temperature T, interacting with the multi-mode transverse field. The kinetic equation for the time-dependent characteristic function

xcu, t) in the linear approximation is (cf. a one-mode case in ref. [13]):

337

Volume 137, number 7,8

PHYSICS LETTERS A

[~ +

(hereregions, Wk=WI and index j is omitted). In transparency where I Wmn W! I >> Ymn, v = 0 and so 0 is unitary. Consider next a Hermitian form ~ ~ ~.v.~*2h_2

i~~w(t)J’—~-+ N(T).~*)

_~(~ + /I.N(T)).W+(t).p*]X(p,

29 May 1989

—

t)=0,

I

(20) where

X ~ Wk

w,.,(t)=w,.,exp[—i(wk—w,)t] M Wk,= ~ ~

2

,

Xak(dmn~Ck)exp(ik~Jj)~

exp[i(k—l)~rJ+i(wk—w,)tOJ

10k

.

1= I

w~~=~

[(amfl—ak)2+y~flfl]’

I

,

(23)

where v=i(w—w~)and >~means the summation over Iki /k. From (23) follows, that ~fi~pnm>~0 (i.e. T>_’ +0), then the matrix v is nonnegative; this in

d%~.ck)(~IO)~* “mn ‘1’ h (w~—w,+iy~)

Ck = ek( hwk)’ /2/2~ (the quantization length is 2~),

is the polarization vector, ~ d~, w$,~and are the relative population difference, the dipole matrix element, the frequency and the phase relaxation parameter ofthe transition rn—n in the jth atom. The matrix wj~~ describes the scattering of the mode / into mode k by the jth atom. Eq. (20) has the following solution es,,

xCu, t)=~(b(t)~p, to)XrCu, t)

,

(21)

turn leads to the nonnegativity of A and to (11). In conclusion the presented TM-formalism gives a compact general method of quantum description of linear optical devices. In some cases it is applicable (with some modifications) also to nonlinear devices [91.

References

where AT= N(T). [I—D(t)~D(t)~

D(t)=exp(_i

J

J

w(t’) dt’)

[1] L. Knoll, W. Vogel and D.G. Welsch, Phys. Rev. A 36 (1987) 3803. [2] B. Huttner and Y. Ben-Aryeh, Phys. Rev. A 38 (1988) 204. [3] 5. Prasad, M.O. Scully andW. Martienssen, Opt. Commun. 62 (1987) 139.

,

[4]H. Fearn and R. Loudon, Opt. Commun. 64 (1987) 485. [5] Z.Y. Ou, C.K. Hong and L. Mandel, Opt. Commun. 63 (1987) 118. [6] A.L. La Porta, R.E. Slusher and B. Yurke, Phys. Rev. Lett.

.

to

Eq. (21) in the limit

t

0—s —00 and t—soo describes the transformation of the input field’s statistics by the atoms of the filter. Thus identifying xCu, —co) =x10Cu), X(/’, 00) X0~,CU)and D(oo) =D we get eq. (6). Note, that

62 (1989) 28. [7] Y.H. Shih and C.O. Alley, Phys. Rev. Lett. 61(1988) 2921; Z.Y. Ou, C.K. Hong and L. Mandel, Opt. Commun. 67 (1988) 159; M.J. Potasek and B. Yurke, Phys. Rev. A 38 P. Grangier, (1988) 3132. [8] M. Ley and R. Loudon, Opt. Commun. 54 (1985) 317.

J

w,.,( t) dt = 27t w,.1ô(w,.

—

w,)

—

i.e. Dk,, which describes the multiple elastic scattering, is diagonal in 1k I = (Ok/C. Consider the matrix

= h.~ m>n v

338

2Yrnn 4Onm (do,,,~ C,.) (dmn .c,)* h2r(w~)2+y2]

(22)

[9] D.N. Klyshko, Photons and nonlinear optics (Gordon and Breach, New York, 1988). [10] M. Born and E. Wolf, Principles of optics (Pergamon, 1964). Soy. Phys. JETP 56 (1982) 753; Zh. Eksp. [11] Oxford, D.N. Klyshko, Teor. Fiz. 94 (1988) 82. [12] D.N. Klyshko, Phys. Lett. A 128 (1988) 133; 132 (1988) 299; Soy. Phys. Usp. 31 (1988) 74; Usp. Fiz. Nauk 158, No.2 (1989). [13]S.E.B. Carusotto, Phys. Rev. A 11(1975)1629; Rockower, N.B. Abraham and S.R. Smith, Phys. Rev. A17(l978)lI00.